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J. Am. Chem. Soc. 2000, 122, 2934-2937
Synthetic Adenine Receptors: Direct Calculation of Binding Affinity and Entropy Ray Luo† and Michael K. Gilson* Contribution from the Center for AdVanced Research in Biotechnology, 9600 Gudelsky DriVe, RockVille, Maryland 20850 ReceiVed NoVember 16, 1999
Abstract: A novel method for calculating binding free energies is applied to a series of water-soluble adenine receptors that have been characterized experimentally. The calculations use a predominant states method, “Mining Minima”, to identify and account for the low-energy conformations of the free and bound species. The CHARMM force field is used to estimate potential energies, and an adjusted form of the generalized Born/ surface area model is used to estimate solvation energies as a function of conformation. The computed binding free energies agree with experiment to within 2.9 kJ/mol (0.7 kcal/mol) and reproduce observed trends across the series of receptors. Preorganization of two rotatable bonds enhances the calculated affinity of one receptor/ adenine complex by -2.5 kJ/mol (-0.6 kcal/mol), and the change in translational/rotational entropy (-T∆S°trans/rot) is 30 kJ/mol (7 kcal/mol). The concept of the translational/rotational entropy change upon binding in the present model is compared with others previously presented in the literature.
Introduction Although biomacromolecules present arguably the most impressive examples of molecular recognition, smaller hostguest systems are of great interest as well. The association of nonmacromolecules has practical applications in a number of areas, including chemical separations, catalysis, and drug delivery. Because of their relative simplicity, host-guest systems are also useful as test cases for computer models of binding. In tests on macromolecules, the adequacy of the conformational sampling in the calculations is often in question, so the success or failure of a calculation may not reflect the validity of the underlying theory and energy model. In contrast, small molecule systems may be simple enough to permit all relevant molecular conformations to be identified and accounted for in the calculations. We have recently developed an efficient method for the direct calculation of free energies and binding affinities1,2 that has given promising results for a series of simple systems.3-8 This method is applied here to seven synthetic adenine receptors that were developed to explore nucleic acid base-pairing in water.9,10 * Corresponding author: Telephone: (301) 738-6217. Fax: (301) 7386255. E-mail:
[email protected]. † Current address: Department of Pharmaceutical Chemistry, University of California, San Francisco, 513 Panassus Avenue, San Francisco, CA 94143-0446. (1) Gilson, M. K.; Given, J. A.; Head, M. S. Chem. Biol. 1997, 4, 8792. (2) Head, M. S.; Given, J. A.; Gilson, M. K. J. Phys. Chem. 1997, 101, 1609-1618. (3) Luo, R.; Head, M. S.; Moult, J.; Gilson, M. K. J. Am. Chem. Soc. 1998, 120, 6138-6146. (4) Luo, R.; David, L.; Hung, H.; Devaney, J.; Gilson, M. K. J. Phys. Chem. 1999, 103, 727-736. (5) Luo, R.; Head, M. S.; Given, J. A.; Gilson, M. K. Biophys. Chem. 1999, 78, 183-193. (6) David, L.; Luo, R.; Head, M. S.; Gilson, M. K. J. Phys. Chem. 1999, 103, 1031-1044. (7) Mardis, K. L.; Glemza, A. J.; Brune, B. J.; Payne, G. F.; Gilson, M. K. J. Phys. Chem. B 1999, 103, 9879-9887. (8) Mardis, K.; Luo, R.; David, L.; Potter, M.; Glemza, A.; Payne, G.; Gilson, M. K. J. Biomolec. Struct. Dyn., in press.
Figure 1. Seven synthetic adenine receptors. The ovals represent the rotatable bonds discussed in the text.
The receptors are equipped with imide moieties that can form hydrogen bonds with adenine and with a variety of different aromatic groups that permit stacking (Figure 1). The calculations yield standard free energies of binding (absolute binding free energies11), permitting direct comparison with measured binding affinities. The calculations also allow issues of preorganization and entropy to be addressed in the context of well-defined theory. Thus, the influence of rigidifying rotatable bonds on (9) Rotello, V. M.; Viani, E. A.; Deslongchamps, G.; Murray, B. A.; Rebek, J., Jr. J. Am. Chem. Soc. 1993, 115, 797-798. (10) Kato, Y.; Conn, M. M.; Rebek, J., Jr. Proc. Natl. Acad. Sci. USA 1995, 92, 1208-1212. (11) Jorgensen, W. L.; Buckner, J. K.; Boudon, S.; Tirado-Rives, J. J. Chem. Phys. 1988, 89, 3742-3746.
10.1021/ja994034m CCC: $19.00 © 2000 American Chemical Society Published on Web 03/08/2000
Binding Affinity and Entropy of Synthetic Adenine Receptors
J. Am. Chem. Soc., Vol. 122, No. 12, 2000 2935
the binding affinity is examined, and changes in translational/ rotational entropy are calculated and compared with values from previous publications. The relationship of the present definition the translational/rotational entropy change on binding is related to the Sackur-Tetrode equation12,13 and the concept of “cratic” entropy.14-17 Methods Calculation of Binding Free Energy. The “Mining Minima” (MM) algorithm is used to compute the aqueous binding affinities of adenine with the seven receptors illustrated in Figure 1. Detailed descriptions of the algorithm and its use have been presented elsewhere.2,4,5 Briefly, the standard free energy change ∆G° when receptor R and ligand L form the noncovalent complex R‚L is given by18
ZR‚L 8π2 ∆G° ) -RT ln + RT ln ZR ZL C°
(1)
where ZX is the configuration integral of the subscripted species, C° is the standard concentration (typically 1 mol/L), and RT is thermal energy. The second term derives from the three external rotations and translations that are converted into internal degrees of freedom in the R‚L complex. The configuration integrals are of the form
ZX ≡
∫ exp(-E(r)/kT)dr
(2)
where E(r) is the energy as a function of the conformation r and kT is thermal energy. As discussed later, the energy E(r) is computed as a sum of potential and solvation energy. The MM algorithm computes the configuration integrals ZX by identifying each low-energy conformation i and evaluating the configuration integral ZX,i for the corresponding energy well via a Monte Carlo method. The complete configurational integral of species X is then
ZX ≈
∑Z
X,i
(3)
i
The sum over energy minima is extended until the Boltzmann-averaged energy 〈E(r)〉 converges to within a predefined tolerance, as described previously.5 Conformational Sampling. Initial all-atom coordinates for adenine and the receptors were generated with Quanta 97.19 Each molecule was then energy-minimized in an appropriate conformation (vide infra) with the full CHARMM 98 vacuum energy function by the NewtonRaphson method20 in version 26 of CHARMM.21 The minimizations were terminated when the energy gradient changed by
